First, in order to have what you are talking about, you have to have, not a single function, F(x, y), but a family of functions.
To take a simple example, look at x^2+ y^2= R^2, the family of all circles with center at the origin. Differentiating both sides of the equation with respect to x, 2x+ 2y dy/dx= 0 so that dy/dx= -x/y at every point. To be orthogonal to that, a function, y(x), must have derivative equal to the negative of the reciprocal. That is, we want dy/dx= -1/(-x/y)= y/x. That is, the "orthogonal complement" of this family of curves must satisfy dy/dx= y/x.
That is an easily separable equation- it can be written dy/y= dx/x. Integrating both sides, ln(|y|)= ln(|x|)+ C so that |y|= C'|x| where C' is equal to the exponential of C. By allowing C to take on non-positive values also, we can drop the absolute values and have y= Cx. That is the family of all straight lines that go through the origin. They are diameters of the original circles so always perpendicular to them.
More generally, given a family of curves, functions of x and y that depend a constant, to find the orthogonal complement, differentiate the equation defining the family to eliminate that constant. Find dy/dx= m(x,y) from that equation and then solve the differential equation dy/dx= -1/m(x,y).
